where \(h=h\,e_3\)  with a uniform and time-independent field strength \(h \in \mathbb {R}\) , and \(\textnormal {m}_3= \langle m, e_3\rangle \) , \(e_3 \in \mathbb {S}^2\) . This effective field \(h_{\textnormal {eff}}\)  also includes the diffusive exchange term \(\partial _x^2m\) , the uniaxial anisotropy and demagnetization field. The specific here with parameter \(\mu \in \mathbb {R}\)  derives from a first order approximation in the thin film/wire limit for a uniformly magnetized body [1]}, [2]}. In the axially symmetric structure, \(\beta \ge 0\)  and \(c_{\textnormal {cp}}\in (-1,1)\)  describe the strength of the spin-transfer and the ratio of the polarization [3]}, [4]}. The spin-transfer torque term may provide energy to the system under certain conditions and counterbalance dissipation associated to the Gilbert damping term, which gives rise to coherent non-variational dynamics, see e.g. [5]}.
